Yes puppies can eat carrots and they are very healthy for your puppy.
Just be sure to cut the carrots up in bite sized pieces so that the puppy can chew and digest the carrots easier.
If you feed the puppy a whole carrot without cutting it up first then the puppy may not chew the carrot all the way and may have trouble digesting the carrot.
However if you cut the carrot up so that it's in bite sized pieces then the puppy can chew it better and digest the carrots better.
You can also feed the puppy other vegetables as well besides carrots but just be sure to cut them up as well to allow the puppy to digest and chew the vegetables much easier.
Most dog foods you buy and puppy foods you buy contain carrots and other vegetables so giving dogs and puppies carrots is perfectly safe and okay.
Your puppies health will benefit when you feed them some carrots and other vegetables.